Summation formulae involving multiple harmonic numbers

Abstract

By means of the generating function approach, we derive several summation formulae involving multiple harmonic numbers Hn,? (?), as well as other combinatorial numbers named after Bernoulli, Euler, Bell, Genocchi and Stirling.

Derivation of computational formulas for certain class of finite sums: Approach to generating functions arising from p$$ p $$‐adic integrals and special functions

“…$$ Similarly, the Euler polynomials and numbers with their generating functions have been extensively studied in mathematics and related areas (see, e.g., cf. other studies 1–32,33–36 ).…”

“…Similarly, generating functions for certain finite sums and their applications have been studied in mathematics and related areas (cf. other studies 1–32,33–36 ).…”

“…A well‐known relationship between the harmonic numbers and the alternating harmonic numbers is $$$ {\mathcal{H}}_n={H}_{\left[\frac{n}{2}\right]}-{H}_n $$$ (cf. other studies 9,11,17,37 ).…”

“…It has very important applications in many fields, especially in mathematics and physics. Using Equation (11), one of the many known special cases of this function is given below (see, e.g., Luke, 23 p. 210, eq. 14 and Srivastava and Choi, 36 p. 67 ):…”

“…The alternating harmonic numbers $$$ {\mathcal{H}}_n $$$ are defined as follows (see, e.g., other studies 4,11,17,36,37 ): $$$$ {\mathcal{H}}_n:= \sum \limits_{j=1}^n\frac{{\left(-1\right)}^j}{j}\kern0.40em \left(n\in \mathbb{N}\right) $$$$ which are generated by means of the following function: $$$$ {F}_2(u)=\frac{\ln \left(1+u\right)}{u-1}=\sum \limits_{n=1}^{\infty }{\mathcal{H}}_n{u}^n\kern0.40em \left(\left|u\right|<1\right). $$$$ …”